Compression properties and constitutive model of short glass fiber reinforced poly-ether-ether-ketone (PEEK)

To analyze the deformation behavior of short glass fiber-reinforced poly-ether-ether-ketone (SGFR-PEEK) under various conditions through numerical simulations, it is crucial to construct a constitutive model that can describe its stress–strain behavior over a wide range of strain rates and temperatures. In this study, quasi-static compression tests were conducted on SGFR-PEEK composites with varying mass fractions, and dynamic tests were performed using a Split Hopkinson Pressure Bar to acquire the material's compressive stress–strain response under quasi-static and dynamic conditions. The results indicate that, under compression, the yield stress of SGFR-PEEK composites increases with an augmentation in glass fiber content, rises with increasing strain rate, and decreases with elevated temperature. Based on experimental findings, a modified Johnson–Cook constitutive model was established to characterize the mechanical performance of SGFR-PEEK. In comparison to the traditional Johnson–Cook intrinsic structure model, the modified model takes into account the glass fiber mass fraction as comprehensively as possible and better predicts the material's flow behavior at high strain rates. Finally, this modified constitutive model was implemented in the ABAQUS software using the user-defined subroutine VUMAT to simulate the compression behavior of SGFR-PEEK composites under different loading conditions, and the model was validated. This research provides valuable insights for the practical application of SGFR-PEEK composites in engineering.

more accurate constitutive models can be constructed to evaluate the influence of strain rate and temperature on material performance.
As a classic phenomenological constitutive model, the Johnson-Cook model for hot-drawn plasticity can consider the effects of strain rate and temperature on plastic stress 17 .Due to its decoupling term and fewer constitutive parameters, the amount of experimental data required to fit its parameters is relatively small, and it is also easier to implement in commercial simulation software.In recent years, quite a few studies based on the Johnson-Cook model and its improved models have achieved good results in describing the mechanical properties of metals and non-metals [18][19][20] .For PEEK materials, Chen et al. 21,22 tested the compression performance of the material under strain rates ranging from 0.01 to 1 s −1 and temperatures ranging from 23 to 150 °C, and, combined with the experimental data from literature 9 , established an improved Johnson-Cook model to describe the mechanical behavior of PEEK, especially pointing out that the yield stress is inversely proportional to temperature.Literature 18 and literature 19 established constitutive models for the material, and, based on custom subroutines, reproduced the studied material's stress-strain in finite element calculation software, successfully simulating the material's response under corresponding application scenarios.This quantitative description of the material's mechanical properties helps to simulate the material's response under different working conditions, making it easier to optimize engineering designs for the material's usage conditions.
In this study, the quasi-static mechanical properties of SGFR-PEEK at different temperatures were tested using a general-purpose materials testing machine with a thermostat box, and the dynamic mechanical properties at different strain rates were obtained using a Hopkinson press bar.The sensitivity of the compressive properties of the material to temperature and strain rate was analyzed, and a modified Johnson-Cook constitutive model taking into account the fiber content was proposed to describe the relationship between stress, strain rate, and temperature.In addition, a user-defined material subroutine, VUMAT, was developed to implement the improved constitutive model in Abaqus ® /Explicit software.The correctness of the improved constitutive model was verified by comparing the experimental results with the simulation results.

Materials and specimens
The SGFR-PEEK composites were produced by Nanjing Comptech ® Composites Corporation (Nanjing, China).These composites consist of fibers with an approximate diameter of 10 μm and lengths ranging from 50 to 200 μm.The melting temperatures (T m ) of various weight fractions of SGFR-PEEK composites were determined through Differential Scanning Calorimetry (DSC) analysis, as summarized in Table 1.It can be seen that with an increase in the glass fiber content, the T m of SGFR-PEEK composites slightly decreases, consistent with the test data in reference 23 .

Quasi-static compression tests
The specimens for quasi-static tests are cylindrical, following the standard GB/T1041-92, with dimensions of ∅10 mm × 15 mm.The experiments were conducted on an electrical universal testing machine, as depicted in Fig. 1.The compression experiments were conducted on an electrically-powered universal testing machine, as   depicted in Fig. 1.This universal testing machine is equipped with a heating box designed for evaluating the mechanical properties of materials under different temperature conditions.The mechanical properties of SGFR-PEEK were tested at room temperature (23 °C) with strain rates of 0.001 s −1 , 0.01 s −1 , 0.1 s −1 , and at elevated temperatures of 50 °C, 100 °C, 150 °C, and 200 °C with a strain rate of 0.001 s −1 .Prior to the variable temperature tests, the specimens were heated and held for 30 min to ensure temperature uniformity.
In the experiments, the engineering stress-engineering strain ( σ e − ε e ) curve could be directly obtained through the data acquisition system, while the true stress-true strain curve ( σ t − ε t ) was calculated using Eq. ( 1).

Dynamic compression tests
The dynamic compression tests are performed on split Hopkinson compression bar apparatus, as shown in Fig. 2, to test the mechanical properties of the material at room temperature (23 °C) with strain rates of 800 s −1 , 1400 s −1 , 2000 s −1 , and 2500 s −1 .The experimental compression rod is a 14.5 mm diameter hard aluminum alloy (LC4) with the elastic modulus of 72 GPa, yield strength of 490 MPa, and density of 2800 kg/m 3 .The lengths of the incident and transmission rods are 1000 mm, and the lengths of the impact and absorption rods are 300 mm.
When conducting dynamic compression tests, achieving a state of stress equilibrium within the specimen requires at least one transmission and reflection of stress waves.To ensure the precision of the experiments, the specimen dimensions were chosen as ∅10 mm × 2 mm in thickness to expedite the time required for the specimen to reach a state of stress equilibrium 24,25 .Additionally, a layer of Vaseline was applied as a lubricant at the points where the specimen contacts the rods to minimize errors resulting from frictional effects.
For data acquisition, the incident and reflected signals were measured using strain gauges affixed to the incident rod, while the transmitted signals were measured using strain gauges mounted on the transmission rod.These strain gauges were positioned at a fixed distance of 500 mm from the specimen.The data acquisition system operated at a sampling frequency of 10 MHz, facilitating real-time visualization of all pulse signals through LabVIEW software.
The SHPB test is based on two basic assumptions: that the stress wave in the rod propagates as a onedimensional elastic wave and that the specimen is uniformly deformed.On the basis of the test waveform, the engineering strain, engineering stress, and strain rate of the specimen can be obtained from Eq. ( 2) [26][27][28][29] .
where ε i , ε r , ε t are the incident, reflected, and transmitted waveforms with time, respectively, L 0 and A 0 are the thickness and cross-sectional area of the specimen, A is the cross-sectional area of the compression bar, and C 0 is the wave speed in the compression bar.A typical set of waveforms in the test is given in Fig. 3.
The true stress-true strain curve of the dynamic compression test is also obtained by Eq. ( 1).

Experimental results
Figure 4 presents the true stress-strain curves of the PEEK matrix and SGFR-PEEK composites with different weight fractions under room temperature (23°C) and quasi-static ( ε=0.001s −1 ) conditions.It can be observed (1) that the SGFR-PEEK composites exhibit an elastic modulus close to that of the PEEK matrix, with a noticeable increase in yield strength.Additionally, as the mass fraction of short glass fibers increases, the yield strength of the composites also rises.In comparison to the yield strength of the PEEK matrix at 134 MPa, the SGFR-PEEK composite material with a 30% weight fraction demonstrates a yield strength of 152 MPa, representing a 13.4% improvement.
Figure 5 illustrates the true stress-strain curves of 30% SGFR-PEEK composites under various testing conditions.Combining this with Fig. 4, it is observed that the stress-strain curves of both the PEEK matrix and SGFR-PEEK composites conform to the four-stage classification outlined in reference 30 : In the first stage (work hardening stage), the work hardening rate is higher than the softening rate caused by dynamic recovery, resulting in a sharp increase in stress at the beginning of micro-strain deformation, followed by an increase in the rate of stress reduction.The second stage (transition stage) is characterized by the competition between dynamic recovery-induced work hardening and softening phenomena and dynamic recrystallization.Additionally, the   www.nature.com/scientificreports/shear stress continues to increase, but at a decreasing rate.In the third stage (softening stage), the stress drops sharply, which is related to dynamic recovery, dynamic recrystallization, and other factors.Finally, in the fourth stage (steady stage), the stress tends to stabilize when a new balance between softening and hardening is achieved.Table 2 presents additional insights into the yield strength of both the PEEK matrix and SGFR-PEEK composites under diverse loading conditions.Notably, the yield strength of SGFR-PEEK composites is influenced significantly by fiber content.Under equivalent conditions, the yield stress increases with higher strain rates and decreases with elevated temperatures.For instance, with a 30% fiber content, the material exhibits a 54.6% enhancement in yield strength at 2500 s −1 compared to 0.001 s −1 at 23 °C.Conversely, under quasi-static loading, the material's yield strength at 200 °C is 77.2% lower than that at 23 °C.These findings underscore the pronounced influence of temperature and strain rate on the mechanical properties of both the PEEK matrix and SGFR-PEEK composites.

Constitutive equations for SGFR + PEEK composites
As an image-only intrinsic structure model, the Johnson-Cook intrinsic structure model has been commonly used for numerical calculations in explosions, impacts, and stamping.In recent years, the JC model and the modified JC model were used to describe the stress changes in PEEK at different temperatures and strain rates with good results 21,22,31 .
Based on the true stress-true strain curves obtained from quasi-static and dynamic tests, each parameter in the JC model can be determined.
The traditional phenomenological JC model may be expressed as: where σ σ is the equivalent flow stress, ε p is the equivalent plastic strain, A is the yield stress at the reference tem- perature and reference strain rate, B is the coefficient of strain hardening, n is the strain hardening exponent, C and m are the material constants which represent the coefficient of strain rate hardening and thermal softening exponent, ε is the strain rate, while εref is the reference strain rate.T * is the homologous temperature and is expressed as: where T is temperature.T ref is the reference temperature.Tm is the melting temperature.

Constitutive equations for 30%SGFR + PEEK
To obtain the parameters of the classical Johnson-Cook intrinsic model, 0.001s −1 was used as the reference strain rate ( εref ) and 20 °C as the reference temperature (T ref ).A is the yield stress of the material at the reference strain rate and reference temperature, which can be determined as 152.08 MPa.T m is the melting temperature, 340.83 °C.At the reference strain rate and reference temperature, Eq. ( 3) can be written as Yield stress of PEEK 9 and SGFR-PEEK under diverse loading conditions.Taking the logarithm of both sides of the equation yields, the values of n and B can be determined as 0.079 and 14.585 MPa.At the reference strain rate, Eq. ( 5) can be written as: the C value obtained using the straight-line fit is 0.0412.Taking the logarithm of both sides, taking the logarithm of both sides of Eq. ( 8), The value of m can be determined from the slope of the average fitted line as 0.962.The classical Johnson-Cook model with all parameters determined by the above calculations can be written as: The comparison between the predicted results of the classical Johnson-Cook model and the experimental results for the 30% SGFR-PEEK is shown in Fig. 6.It is evident that the predicted values of the classical model are in good agreement with the experimental values at low strain rates and various temperatures.However, at high strain rates, there are significant discrepancies between the predicted and experimental values, and the classical model is unable to describe the material's deformation stages of work hardening, transition, and softening.This indicates that the classical Johnson-Cook model cannot fully reflect the mechanical behavior of the material over a wide range of strain rates and temperatures.To better describe the material's mechanical properties, it is necessary to improve the Johnson-Cook model.
To provide a more accurate description of the mechanical properties of materials, several studies have proposed methods to improve the JC model.These approaches often involve rewriting and refitting certain terms within the conventional JC model using polynomial or power functions, aiming to achieve better performance 21,22 .In this study, we followed the strategy outlined in these previous investigations and developed a modified JC model to better characterize the mechanical behavior of SGFR-PEEK composites.
Similarly, 0.001 s −1 was used as the reference strain rate εref and 23 °C as the reference temperature T ref to determine the parameters associated with the modified Johnson-Cook model.
To enable the improved model to describe the behavior of materials such as process hardening, the first term (A + Bε n ) in the classical Johnson-Cook model is rewritten in polynomial form as: The fitted values of A and B 1 ~ B 6 are given in Table 3.To express the influence of strain rate on material properties, a function is constructed to fit the relationship between σ/σ(ε p ) and ln(ε/ε ref ).( 6) have values that are relatively close across various models.This suggests that changes in fiberglass content alter the specific characteristics of the stress-strain curve but do not significantly impact the material's patterns of strain rate strengthening and temperature softening.Therefore, in practical applications, to obtain an accurate stress-strain relationship for the material, it is necessary to experimentally determine the specific values of B 1 to B 6 .Meanwhile, the specific values of A, C 1 , C 2 , and m can be referenced from the computational results presented in this paper.
Following the above analysis, the modified Johnson-Cook model for SGFR-PEEK composites, considering fiber content, can be expressed as follows: where w represents the mass fraction of fiberglass, the values of B 1 to B 6 can be experimentally determined.Additionally, C 1 = 0.02604, C 2 = 9.58 × 10 -4 , m = 0.969.
Table 5 illustrates the error between the results calculated from Eq. ( 15) and experimental results.It can be observed that under conditions ranging from 23°C to 100°C, the error between calculated values and experimental results is generally within 10%.However, at temperatures of 150°C and 200°C, the model exhibits larger (15)

Finite element calculation
In order to achieve more accurate numerical calculations of SGFR-PEEK composites under various loading conditions to assist engineering practice, a user-defined material subroutine VUMAT was developed based on the modified Johnson-Cook model.The VUMAT subroutine can be used for explicit dynamic calculations, and its fixed-format interface ensures that it can be called by the ABAQUS main program.The flowchart of the VUMAT and ABAQUS analysis process is shown in Fig. 8.
The VUMAT subroutine includes the parameters required by the improved Johnson-Cook model, as well as the material's elastic modulus and Poisson's ratio, which are 2800 MPa and 0.31, respectively.This paper established a finite element model consistent with the quasi-static experimental dimensions, as shown in Fig. 9.The material parameters of the specimen were defined by the VUMAT subroutine, while the compression platens and base were made of steel with an elastic modulus of 210 GPa and Poisson's ratio of 0.3, without considering their plastic deformation.The friction coefficient between the specimen and the platens is 0.2.During the simulation, the base was fixed, and the compression platens moved downward with a velocity set according to the experimental reference.The model mesh used eight-node hexahedral reduced integration elements (C3D8R), and the mesh of the specimen was refined, resulting in a total of 94,800 elements.
The stress-strain curves of SGFR-PEEK composites with different mass fractions obtained from numerical simulations and experiments under various conditions are illustrated in Fig. 10.It is evident that within the plastic range, the results of numerical simulations under different conditions closely align with experimental outcomes.This indicates that users can simulate the mechanical properties of SGFR-PEEK composites with varying component mass fractions by defining necessary parameters in VUMAT.Furthermore, more sophisticated simulation calculations can be conducted to assist in simulation design.
Finally, it should be noted that, referring to the tensile properties of the material in various literature sources 9,34,35 , as illustrated in Fig. 11, the material exhibits different yield strengths during tension and compression.This implies that, despite the modified JC model and VUMAT subroutine have shown good performance in describing the compression performance of materials, significant deviations are still observed when characterizing the material under tensile stress.Achieving a comprehensive understanding of the material's tensile and compressive performance requires further experimentation and analysis in subsequent studies.

Conclusion
Through quasi-static tests at different temperatures and Split Hopkinson Pressure Bar (SHPB) compression tests at room temperature, the mechanical properties of short glass fiber-reinforced poly-ether-ether-ketone (SGFR-PEEK) composites with varying mass fractions were investigated under different temperatures and strain rates.The results indicate that the yield stress of the material increases with an augmentation in the glass fiber content, demonstrating sensitivity to both temperature and strain rate.Specifically, the yield stress decreases with an increase in temperature but increases with higher strain rates.A modified Johnson-Cook model was established based on experimental findings to describe the material's mechanical behavior.This model incorporates www.nature.com/scientificreports/glass fiber mass fraction and effectively captures the dependency of the material's mechanical performance on temperature and strain rate.Furthermore, the enhanced constitutive model was implemented in ABAQUS software using the VUMAT subroutine to simulate SGFR-PEEK composite behavior.The accuracy of the developed constitutive model was validated through a comparison of simulation results with experimental data.The research findings contribute valuable insights for the engineering applications and numerical simulations of SGFR-PEEK composites.

Table 4 .
18 + 58.91 w 100% The values of each parameter in the modified Johnson-Cook model.

Table 5 .
The error between the improved Johnson-Cook model and experimental results.